Question: $ A = \left[\begin{array}{rrr}1 & -2 & 2 \\ 0 & -2 & 3\end{array}\right]$ $ B = \left[\begin{array}{rr}2 & 4 \\ 2 & 4 \\ -2 & 1\end{array}\right]$ What is $ A B$ ?
Explanation: Because $ A$ has dimensions $(2\times3)$ and $ B$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ A B = \left[\begin{array}{rrr}{1} & {-2} & {2} \\ {0} & {-2} & {3}\end{array}\right] \left[\begin{array}{rr}{2} & \color{#DF0030}{4} \\ {2} & \color{#DF0030}{4} \\ {-2} & \color{#DF0030}{1}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ B$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ B$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ B$ , and so on. Add the products together. $ \left[\begin{array}{rr}{1}\cdot{2}+{-2}\cdot{2}+{2}\cdot{-2} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{1}\cdot{2}+{-2}\cdot{2}+{2}\cdot{-2} & ? \\ {0}\cdot{2}+{-2}\cdot{2}+{3}\cdot{-2} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ A$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{1}\cdot{2}+{-2}\cdot{2}+{2}\cdot{-2} & {1}\cdot\color{#DF0030}{4}+{-2}\cdot\color{#DF0030}{4}+{2}\cdot\color{#DF0030}{1} \\ {0}\cdot{2}+{-2}\cdot{2}+{3}\cdot{-2} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{1}\cdot{2}+{-2}\cdot{2}+{2}\cdot{-2} & {1}\cdot\color{#DF0030}{4}+{-2}\cdot\color{#DF0030}{4}+{2}\cdot\color{#DF0030}{1} \\ {0}\cdot{2}+{-2}\cdot{2}+{3}\cdot{-2} & {0}\cdot\color{#DF0030}{4}+{-2}\cdot\color{#DF0030}{4}+{3}\cdot\color{#DF0030}{1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}-6 & -2 \\ -10 & -5\end{array}\right] $